Unsupervised Discovery of Formulas for Mathematical Constants

Ongoing efforts that span over decades show a rise of AI methods for accelerating scientific discovery, yet accelerating discovery in mathematics remains a persistent challenge for AI. Specifically, AI methods were not effective in creation of formulas for mathematical constants because each such formula must be correct for infinite digits of precision, with “near-true” formulas providing no insight toward the correct ones. Consequently, formula discovery lacks a clear distance metric needed to guide automated discovery in this realm. In this work, we propose a systematic methodology for categorization, characterization, and pattern identification of such formulas. The key to our methodology is introducing metrics based on the convergence dynamics of the formulas, rather than on the numerical value of the formula. These metrics enable the first automated clustering of mathematical formulas. We demonstrate this methodology on Polynomial Continued Fraction formulas, which are ubiquitous in their intrinsic connections to mathematical constants, and generalize many mathematical functions and structures. We test our methodology on a set of 1,768,900 such formulas, identifying many known formulas for mathematical constants, and discover previously unknown formulas for $π$, $\ln(2)$, Gauss’, and Lemniscate’s constants. The uncovered patterns enable a direct generalization of individual formulas to infinite families, unveiling rich mathematical structures. This success paves the way towards a generative model that creates formulas fulfilling specified mathematical properties, accelerating the rate of discovery of useful formulas.

💡 Research Summary

The paper tackles the long‑standing problem of automatically discovering closed‑form formulas for mathematical constants, a task traditionally reliant on deep human intuition because a formula must be exact for infinitely many digits. The authors focus on polynomial continued fractions (PCFs), which are defined by two integer‑coefficient polynomials aₙ and bₙ and encompass a wide variety of constants and functions (e.g., π, ln 2, Bessel functions).

First, they generate all possible PCFs with quadratic polynomials whose coefficients lie in the range –5 to 5, yielding 1,768,900 candidate formulas. After discarding degenerate cases (a=0 or b=0) and filtering out non‑convergent fractions using a custom convergence classifier (Appendix B), they retain 1,543,926 convergent PCFs.

The core contribution is a set of dynamics‑based metrics that describe the behavior of the rational approximants pₙ/qₙ generated by each PCF, rather than the coefficients of the defining polynomials or the numerical limit itself. The three main metric families are:

1. Irrationality measure δ – traditionally defined via the limit of –log|L−pₙ/qₙ| / log qₙ, which requires knowledge of the limit L. The authors introduce two tools to estimate δ without L: (a) the Blind‑δ algorithm, which infers δ directly from the sequence of approximants, and (b) the δ‑Predictor formula (Eq. 5) that uses the dominant eigenvalues λ₁, λ₂ of the 2×2 matrix